A245687 Number T(n,k) of endofunctions on [n] such that the minimal cardinality of the nonempty preimages equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 24, 0, 3, 0, 216, 36, 0, 4, 0, 2920, 200, 0, 0, 5, 0, 44100, 2250, 300, 0, 0, 6, 0, 799134, 22932, 1470, 0, 0, 0, 7, 0, 16429504, 342608, 3136, 1960, 0, 0, 0, 8, 0, 382625856, 4638384, 147168, 9072, 0, 0, 0, 0, 9, 0, 9918836100, 79610850, 1522800, 18900, 11340, 0, 0, 0, 0, 10

Offset

0,5

Comments

T(0,0) = 1 by convention.

Links

Alois P. Heinz, Rows n = 0..100, flattened

Example

Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,      2;
  0,       24,      0,    3;
  0,      216,     36,    0,    4;
  0,     2920,    200,    0,    0,  5;
  0,    44100,   2250,  300,    0,  0,  6;
  0,   799134,  22932, 1470,    0,  0,  0,  7;
  0, 16429504, 342608, 3136, 1960,  0,  0,  0,  8;
  ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-j, i-1, k)/j!, j=k..n)))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), `if`(k=n, n,
    `if`(k>=(n+1)/2, 0, n!*(b(n$2, k)-b(n$2, k+1))))):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-j, i-1, k]/j!, {j, k, n}]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], If[k == n, n, If[k >= (n+1)/2, 0, n!*(b[n, n, k] - b[n, n, k+1])]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 02 2015, after Alois P. Heinz *)

Crossrefs

T(n,1) = n*A241581(n) for n>0.

Rows sums give A000312.

Main diagonal gives A028310.

T(2n,n) gives A273325.

Cf. A019575 (the same for maximal cardinality).

Sequence in context: A151339 A228273 A069521 * A228617 A119836 A158112

Adjacent sequences: A245684 A245685 A245686 * A245688 A245689 A245690

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 29 2014

Status

approved